Measuring Distances With Geometry

We can convert baselines and parallaxes into distances using arguments made by the Greek geometer Euclid. The figure below represents Figure 1.25(a), but we have changed the scale and added the circle centered on the target planet and passing through our baseline on Earth.

To compute the planet’s distance, we note that the ratio of the baseline AB to the circumference of the large circle shown in the figure must be equal to the ratio of the parallax to one full revolution, 360°. Since the radius of the large circle is 2π times the distance to the planet (where π—the Greek letter "pi"—is approximately equal to 3.142), it follows that

from which we find

The angle 360°/2π 57. 3° in the above equation is usually called 1 radian.

EXAMPLE: Two observers 1000 km apart looking at the Moon might measure (using the photographic technique described in the text) a parallax of 9.0 arc minutes—that is, 0.15°. It then follows that the distance to the Moon is 1000 km x (57.3/0.15) 380,000 km. (More accurate measurements, based on laser ranging using equipment left on the lunar surface by Apollo astronauts, yield a mean distance of 384,000 km.)

Knowing the distance to an object, we can then determine many other properties. For example, by measuring the object’s angular diameter, we can compute its size. The figure below illustrates the geometry involved.

Notice that this is basically the same diagram as the previous one, except that now the angle (the angular diameter) and distance are known, instead of the angle (the parallax) and baseline. The same reasoning as before then allows us to calculate the diameter:

so

EXAMPLE: The Moon’s angular diameter is measured to be about 31 arc minutes—a little over half a degree. From the preceding discussion, it follows that the Moon’s actual diameter is 380,000 km x (0.52°/57.3°) 3450 km. A more precise measurement gives 3476 km.

Study the above reasoning carefully. Simple measurements such as these form the basis for almost every statement made in this book about size and scale in the universe.